We mention that () is imposed even for asymptotic analysis of ASGD [1]}.
It ensures the Lipschitz continuity of the mapping \({x}\rightarrow \mathbb {E}[\nabla f({x}; \xi )\nabla ^T f({x}; \xi )]\)  (as proved in (REF )).
See [1]} for the discussion.
We note that [3]} directly assumed the mapping is continuous.
We prefer to adopt () due to two reasons.
First, () has a clear connection to ().
It is satisfied by a variety of objective functions, such as least squares regression \(f({x};\xi ) = (\xi _y - {x}^T\xi _{{x}})^2/2\)  and logistic regression \(f({x}; \xi ) = \log {1+\exp {{x}^T\xi _{{x}}}} - \xi _y\cdot {x}^T\xi _{{x}}\) , where \(\xi = (\xi _y, \xi _{{x}})\)  is the response-feature pair with \(\xi _y\in \mathbb {R}, \xi _{{x}}\in \mathbb {R}^d\)  for least squares regression and \(\xi _y\in \lbrace 0, 1\rbrace , \xi _{{x}}\in \mathbb {R}^d\)  for logistic regression, as long as \(\xi _{{x}}\)  has bounded 4-th moment.
This condition is more intuitive and checkable than the continuity of the mapping.
Second, although our asymptotic normality and Berry-Esseen bound also hold under the continuity condition on the mapping, () further enables us to design a practical covariance estimator with an explicit convergence rate (cf. Lemma REF ).
The covariance matrix estimation is not considered in [3]}.
